Geometry of the eigencurve at critical Eisenstein series of weight 2

Majumdar, Dipramit

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 1, pp. 183-197.

Dans cet article, nous montrons que la série d’Eisenstein critique de poids 2, $E_2^{cri{t}_p}$, définit un point lisse dans la courbe de Hecke $ℂ\left(l\right)$, où $l$ est un nombre premier différent de $p$. Nous montrons également que $E_2^{cri{t}_p,or{d}_l}$ définit un point lisse dans la courbe de Hecke pleine ${ℂ}^{full}\left(l\right)$ et que le point défini par $E_2^{cri{t}_p,or{d}_{l_1},or{d}_{l_2}}$ est non lisse dans la courbe de Hecke pleine ${ℂ}^{full}\left(l_1{l}_2\right)$. En outre, nous montrons que $ℂ\left(l\right)$ est étale sur l’espace des poids au point défini par $E_2^{cri{t}_p}$. En conséquence, nous montrons que la conjecture d’abaissement du niveau de Paulin n’est pas valide pour $E_2^{cri{t}_p,or{d}_l}$.

In this paper we show that the critical Eisenstein series of weight 2, $E_2^{cri{t}_p}$, defines a smooth point in the eigencurve $ℂ\left(l\right)$, where $l$ is a prime different from $p$. We also show that $E_2^{cri{t}_p,or{d}_l}$ defines a smooth point in the full eigencurve ${ℂ}^{full}\left(l\right)$ and $E_2^{cri{t}_p,or{d}_{l_1},or{d}_{l_2}}$ defines a non-smooth point in the full eigencurve ${ℂ}^{full}\left(l_1{l}_2\right)$. Further, we show that $ℂ\left(l\right)$ is étale over the weight space at the point defined by $E_2^{cri{t}_p}$. As a consequence, we show that level lowering conjecture of Paulin fails to hold at $E_2^{cri{t}_p,or{d}_l}$.

Classification : 11F85, 11F11, 11F80

[1] J. Bellaïche, Critical p-adic L-functions. Invent. Math.189, (2012), 1–60. | MR

[2] J. Bellaïche, Introduction to the Conjecture of Bloch and Kato. Available at http://people.brandeis.edu/~jbellaic/BKHawaii5.pdf.

[3] J. Bellaïche, G. Chenevier, Lissité de la Courbe de Hecke de $\mathrm{GL}\_2$ aux Points Eisenstein Critiques. J. Inst. Math. Jussieu.5, (2006), 333–349. | MR | Zbl

[4] J. Bellaïche, G. Chenevier, Families of Galois representations and Selmer groups. Astérisque.324, (2009), xii+314. | MR | Zbl

[5] J. Bellaïche, , G. Chenevier, Sous-groupes de ${\mathrm{GL}}_2$ et arbres, J. Algebra.410, (2014),501–525. | MR | Zbl

[6] J.Bellaïche, M. Dimitrov, On the Eigencurve at classical weight one points. ArXiv e-prints (2013).

[7] K.Buzzard, Eigenvarieties. L-functions and Galois representations, London Math. Soc. Lecture Note Ser.320 (2007),59–120. | MR | Zbl

[8] R.Coleman, B.Mazur, The eigencurve. Galois representations in arithmetic algebraic geometry(Durham, 1996), London Math. Soc. Lecture Note Ser.254 (1998),1–113. | MR | Zbl

[9] R. Coleman, F. Q. Gouvêa, N. Jochnowitz, $E_2$, $\Theta$, and overconvergence. Int. Math. Res. Not. (1995), 23–41. | MR | Zbl

[10] M. Dimitrov, E. Ghate, On classical weight one forms in Hida families. J. Théor. Nombres Bordeaux. 24 (2012), 669–690. | Numdam | MR | Zbl

[11] H. Hida, Galois representations into ${\mathrm{GL}}_2\left({\mathbf{Z}}_p\left[\left[X\right]\right]\right)$ attached to ordinary cusp forms. Invent. Math. 85 (1986), 545–613. | MR | Zbl

[12] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture. Invent. Math. 153 (2003), 373–454. | MR | Zbl

[13] N. Koblitz, $2$-adic and $3$-adic ordinals of $(1/j)$-expansion coefficients for the weight $2$ Eisenstein series. Bull. London Math. Soc. 9 (1977), 188–192. | MR | Zbl

[14] T. Miyake, Modular Forms. Springer-Verlag, Berlin. (1989), x+335, translated from the Japanese by Yoshitaka Maeda. | MR | Zbl

[15] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 323 (2008), xvi+825. | MR | Zbl

[16] A. G. M. Paulin, Local to global compatibility on the eigencurve. Proc. Lond. Math. Soc. (3) 103 (2011), 405–440. | MR | Zbl

[17] A. G. M. Paulin, Geometric level raising and lowering on the eigencurve. Manuscripta Math. 137 (2012), 129–157. | MR | Zbl

[18] C. Soulé, On higher $p$-adic regulators. Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., (1980), Lecture Notes in Math. 854 (1981), 372–401. | MR | Zbl

[19] W. A. Stein, Modular Forms, A Computational Approach. Book available at http://modular.math.washington.edu/books/modform/modform/index.html. | MR

[20] R. Taylor, Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63 (1991), 281–332. | MR | Zbl